The authors prove a series of results about uniform \(s\)-boundedness and absolute continuity, a measure extension theorem and a result generalizing the Vitali-Hahn-Saks theorem for measures with values in Dedekind complete weakly distributive lattice groups. A crucial idea in this paper is to use in the definitions of \(s\)-boundedness, \(\sigma\)-additivity and convergence of measures, a special form of convergence in lattice groups called by the authors RD-convergence which is defined by means of a regulator \((a_{ij})_{i,j\in\mathbb{N}}\).